On the symmetries of some classes of recursive circulant graphsSeyed MortezaMirafzalLorestan Universityauthortextarticle2014engA recursive-circulant $G(n; d)$ is defined to be a‎ ‎circulant graph with $n$ vertices and jumps of powers of $d$‎. ‎$G(n; d)$ is vertex-transitive‎, ‎and has some strong hamiltonian‎ ‎properties‎. ‎$G(n;d)$ has a recursive structure when $n = cd^m$‎, ‎$1 \leq c < d $ [Theoret‎. ‎Comput‎. ‎Sci. 244 (2000) 35-62]‎. ‎In this paper‎, ‎we will find the automorphism‎ ‎group of some classes of recursive-circulant graphs‎. ‎In particular‎, ‎we‎ ‎will find that the automorphism group of $G(2^m; 4)$ is isomorphic‎ ‎with the group $D_{2 \cdot 2^m}$‎, ‎the dihedral group of order $2^{m+1}$‎.Transactions on CombinatoricsUniversity of Isfahan2251-86573

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201416http://toc.ui.ac.ir/article_3818_b3c38ea56007a423479c7e20e97949c7.pdfdx.doi.org/10.22108/toc.2014.3818On the number of mutually disjoint cyclic designsMojganEmamiDepartment of Mathematics, University of ZanjanauthorOzraNaserianDepartment of Mathematics, University of Zanjanauthortextarticle2014engWe denote by $LS[N](t,k,v)$ a large set of $t$-$(v,k,\lambda)$ designs of size $N$‎, ‎which is a partition of all $k$-subsets of‎ ‎a $v$-set into $N$ disjoint $t$-$(v,k,\lambda)$ designs and‎ ‎$N={v-t \choose k-t}/\lambda$‎. ‎We use the notation‎ ‎$N(t,v,k,\lambda)$ as the maximum possible number of mutually‎ ‎disjoint cyclic $t$-$(v,k,\lambda)$designs‎. ‎In this paper we give‎ ‎some new bounds for $N(2,29,4,3)$ and $N(2,31,4,2)$‎. ‎Consequently‎ ‎we present new large sets $LS[9](2,4,29)‎, ‎LS[13](2,4,29)$ and‎ ‎$LS[7](2,4,31)$‎, ‎where their existences were already known‎.Transactions on CombinatoricsUniversity of Isfahan2251-86573

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2014713http://toc.ui.ac.ir/article_3820_cf20d0b9a831a25a6c55e7bf6461ca78.pdfdx.doi.org/10.22108/toc.2014.3820Some designs and codes from $L_2(q)$JamshidMooriNorth-West University (Mafikeng Campus)authorGeorgesRandriafanomezantsoaNorth-West Universityauthortextarticle2014eng‎For $q \in \{7,8,9,11,13,16\}$‎, ‎we consider the primitive actions of $L_2(q)$ and use Key-Moori Method 1 as described in [Codes‎, ‎designs and graphs from the Janko groups {$J_1$} and‎ ‎{$J_2$}‎, J‎. ‎Combin‎. ‎Math‎. ‎Combin‎. ‎Comput.‎, ‎40 (2002) 143-159.‎, ‎Correction to‎: ‎``Codes‎, ‎designs and graphs from the Janko groups‎ ‎{$J_1$} and {$J_2$}'' [J‎. ‎Combin‎. ‎Math‎. ‎Combin‎. ‎Comput‎., 40 (2002) 143-159]‎, J‎. ‎Combin‎. ‎Math‎. ‎Combin‎. ‎Comput.‎, 64} (2008) 153.] to construct designs from the orbits of the point stabilisers and from any union of these orbits‎. ‎We also use Key-Moori Method 2‎ ‎(see Information security‎, ‎coding theory and related combinatorics‎, ‎NATO Sci‎. ‎Peace Secur‎. ‎Ser‎. ‎D Inf‎. ‎Commun‎. ‎Secur.‎, ‎IOS Amsterdam‎, 29 (2011) 202-230.) to determine the designs from the maximal subgroups and the conjugacy classes of elements of these groups‎. ‎The incidence matrices of these designs are then used to generate associated binary codes‎. ‎The full automorphisms of these designs and codes are also determined‎.Transactions on CombinatoricsUniversity of Isfahan2251-86573

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20141528http://toc.ui.ac.ir/article_3821_42ec8c804723ec7607f38bcb34d584a7.pdfdx.doi.org/10.22108/toc.2014.3821On the number of maximum independent sets of graphsTajedinDerikvandIslamic Azad University of MarvdashtauthorMohammad RezaOboudiUniversity of Isfahanauthortextarticle2014engLet $G$ be a simple graph‎. ‎An independent set is a set of‎ ‎pairwise non-adjacent vertices‎. ‎The number of vertices in a maximum independent set of $G$ is‎ ‎denoted by $\alpha(G)$‎. ‎In this paper‎, ‎we characterize graphs $G$ with $n$ vertices and with maximum‎ ‎number of maximum independent sets provided that $\alpha(G)\leq 2$ or $\alpha(G)\geq n-3$‎.Transactions on CombinatoricsUniversity of Isfahan2251-86573

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20142936http://toc.ui.ac.ir/article_4060_0f0b72655269c445900a6c2f92064cef.pdfdx.doi.org/10.22108/toc.2014.4060Extremal skew energy of digraphs with no even cyclesJingLiDepartment of Applied Mathematics, Northwestern Polytechnical UniversityauthorXueliangLiCenter for Combinatorics and LPMC-TJKLC, Nankai UniversityauthorHuishuLianCenter for Combinatorics, Nankai Universityauthortextarticle2014eng‎Let $D$ be a digraph with skew-adjacency matrix $S(D)$‎. ‎Then the‎ ‎skew energy of $D$ is defined as the sum of the norms of all‎ ‎eigenvalues of $S(D)$‎. ‎Denote by $\mathcal{O}_n$ the class of‎ ‎digraphs of order $n$ with no even cycles‎, ‎and by‎ ‎$\mathcal{O}_{n,m}$ the class of digraphs in $\mathcal{O}_n$ with‎ ‎$m$ arcs‎. ‎In this paper‎, ‎we first give the minimal skew energy‎ ‎digraphs in $\mathcal{O}_n$ and $\mathcal{O}_{n,m}$ with $n-1\leq‎ ‎m\leq \frac{3}{2}(n-1)$‎. ‎Then we determine the maximal skew energy‎ ‎digraphs in $\mathcal{O}_{n,n}$ and $\mathcal{O}_{n,n+1}$‎, ‎and in‎ ‎the latter case we assume that $n$ is even‎.Transactions on CombinatoricsUniversity of Isfahan2251-86573

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20143749http://toc.ui.ac.ir/article_4059_ed37714018ee8fbea9af32b692471f46.pdfdx.doi.org/10.22108/toc.2014.4059Watching systems of triangular graphsMaryamRoozbayaniScience and Research Branch, Islamic Azad UniversityauthorHamidrezaMaimaniShahid Rajaee Teacher Training UniversityauthorAbolfazlTehranianScience and Research Branch, Islamic Azad Universityauthortextarticle2014engA watching system in a graph $G=(V‎, ‎E)$ is a set $W=\{\omega_{1}‎, ‎\omega_{2}‎, ‎\dots‎, ‎\omega_{k}\}$‎, ‎where $\omega_{i}=(v_{i}‎, ‎Z_{i})‎, ‎v_{i}\in V$ and $Z_{i}$ is a subset of closed neighborhood of‎ ‎$v_{i}$ such that the sets $L_{W}(v)=\{\omega_{i}‎: ‎v\in‎ ‎Z_{i}\}$ are non-empty and distinct‎, ‎for any $v\in V$‎. ‎In this‎ ‎paper‎, ‎we study the watching systems of line graph $K_{n}$ which is‎ ‎called triangular graph and denoted by $T(n)$‎. ‎The minimum size of a‎ ‎watching system of $G$ is denoted by $\omega(G)$‎. ‎We show that‎ ‎$\omega(T(n))=\lceil\frac{2n}{3}\rceil$‎.Transactions on CombinatoricsUniversity of Isfahan2251-86573